Boundary States Boundary States and Black p-branesand Black p-branes

Shinpei Kobayashi Shinpei Kobayashi （（ RESCEURESCEU ））

in collaboration with in collaboration with

Tsuguhiko Asakawa (RIKEN)Tsuguhiko Asakawa (RIKEN)

So Matsuura (RIKEN)So Matsuura (RIKEN)2004/05/19, 関東ゼミ

1. Introduction1. Introduction

How should we apply string theory How should we apply string theory to gravitational systems ? to gravitational systems ?What is ‘string cosmology’ ?What is ‘string cosmology’ ?

→ → D-brane is thought to be a key to D-brane is thought to be a key to describe well-known gravitational systems describe well-known gravitational systems

via string theory. via string theory.

D-braneD-brane

Open string endpoints can stick to Open string endpoints can stick to D-braneD-brane

D-branes carry D-branes carry RR chargesRR charges

)(|:)1,...,1(

)(0|:),...,1,0(

0

0

DirichletxXDpiX

NeumannXpXiii

X0

X Xi

σ=0 σ=π

τ

Dynamical non-BPS D-brane systems are Dynamical non-BPS D-brane systems are very important in string theory, very important in string theory,

(e.g.)(e.g.) D(2p+1)-brane in type IIA stringD(2p+1)-brane in type IIA string D(2p)-brane in type IIB stringD(2p)-brane in type IIB string D/anti D-brane systemD/anti D-brane system

(c.f.) BPS D-branes, (c.f.) BPS D-branes, stablestable non-BPS D-brane non-BPS D-brane

But no one has succeeded in describing But no one has succeeded in describing the dynamics of non-BPS D-brane.the dynamics of non-BPS D-brane.

Non-BPS D-brane system (1)Non-BPS D-brane system (1)D(2p+1)-braneD(2p+1)-brane

Closed string vacuum

Non-BPS D-brane system (2)Non-BPS D-brane system (2)D/anti D-brane systemD/anti D-brane system

closed string vacuum

lower-dimensional D-brane

Importance of dynamical D-brane Importance of dynamical D-brane systemssystems

String theoryString theory Searching for ‘real’ vacuum of string theorySearching for ‘real’ vacuum of string theory String interaction & dynamicsString interaction & dynamics

→ → 　　 non-perturbative string theorynon-perturbative string theory Gravitation & CosmologyGravitation & Cosmology

D-brane inflationD-brane inflation Black hole evaporation Black hole evaporation

　 → 　 → Application to physics at Planck scaleApplication to physics at Planck scale

Trials to dynamical D-brane Trials to dynamical D-brane systemssystems

Via ‘non-perturbative’ string theoryVia ‘non-perturbative’ string theory Open string field theory (A.Sen, …)Open string field theory (A.Sen, …) Closed string field theory Closed string field theory

(Asakawa, SK &Matsuura (’03), …) (Asakawa, SK &Matsuura (’03), …)

Via conformal field theoryVia conformal field theory Logarithmic CFT description Logarithmic CFT description

(Asakawa, Ishimoto, SK & Matsuura, (Asakawa, Ishimoto, SK & Matsuura, work in progress) work in progress)

Trials to dynamical D-brane Trials to dynamical D-brane systemssystems

Via low-energy effective theoryVia low-energy effective theory (Zhou & Zhu (‘99), Ohta & Yokono (‘02)(Zhou & Zhu (‘99), Ohta & Yokono (‘02) Brax, Mandal & Oz (‘01)) Brax, Mandal & Oz (‘01))

Time-dependent solutions have not found yet.Time-dependent solutions have not found yet. Stable BPS solution →Stable BPS solution → 　　 OKOK

black p-branes : black p-branes : Today’s themeToday’s theme Non-BPS solution → ?Non-BPS solution → ?

Dynamical systemHawking radiation,

Inflation, etc.

unknownobject

SUGRA String theory

Unknown non-BPSblack p-brane

(BPS) black p-brane

non-BPS D-brane

BPS D-brane

?

D-brane/black p-brane relationD-brane/black p-brane relation Stable BPS D-brane caseStable BPS D-brane case (Unstable non-BPS case)(Unstable non-BPS case)

Black p-brane Black p-brane from boundary state (= D-brane) from boundary state (= D-brane)

(Difference between D-brane (Difference between D-brane and black p-brane) and black p-brane)

2. Black p-brane 2. Black p-brane

Classical solution of SUGRAClassical solution of SUGRA It has same symmetry, charge and mass It has same symmetry, charge and mass

as a D-braneas a D-brane

→ → Low-energy description of a D-brane.Low-energy description of a D-brane. But no one has provedBut no one has proved..

(Non-BPS black p-branes have not been (Non-BPS black p-branes have not been found yet)found yet)

String Theory and SUGRAString Theory and SUGRA

String Field Theory action

Classical solution ofSting theory

Dp-brane

Classical solution ofSUGRA

Black p-brane

Supergravity actionmassless

massless

EO

M

EO

M

SUGRA action & ansatzSUGRA action & ansatz

11

21

2

)(

,||)!1(2

1

2

1

nn

naD

dAF

Fen

RgxdS

・ Φ ： dilaton ・ A ： n-form potential ・ F ： (n+1)-form field strength

X0

XXi

σ=0 σ=π

τ

jiij dxdxrgdxdxrbds

)()( 222

Black p-brane solutionBlack p-brane solution

.42

)3)(1(2

,1

)3(

21)(

,)()(,)()(

,)()(

2

32

1...01

2

2

1

2

32

aD

pDp

rpD

TrH

where

rHrArHre

dxdxrHdxdxrHds

pDpD

p

p

a

jiij

D

p

D

pD

3. Boundary state3. Boundary state

D-brane in closed string channelD-brane in closed string channel Source of closed stringsSource of closed strings

← Such properties are guaranteed by ← Such properties are guaranteed by conformal symmetry of the world-sheet conformal symmetry of the world-sheet

conformal transformationconformal transformationζ→ ζ→ ff(ζ), where ζ=σ+iτ(ζ), where ζ=σ+iτ

Using the conformal transformation, we can Using the conformal transformation, we can change the boundary condition for open change the boundary condition for open strings into that for closed strings.strings into that for closed strings.

),(),(.

,

iii

)1,...,1(|

),...,1,0(0|

0

0

DpixX

pXii

)1,...,1(|

),...,1,0(0|

0

0

DpixBX

pBXi

Xi

X

Closed string

Boundary state

Closed string tree graph

Open string

D-brane

Open string 1-loop graph

We can rewrite the boundary condition with We can rewrite the boundary condition with using the oscillators.using the oscillators.

).exp(

,)(1

iz

zizXn

nn

0)ˆ(;0ˆ

),(,0)~(

,0,0)~(;0)~(

Xii

X

ijXnn

Xin

inXnn

BxqBp

SBS

nBB

000)ˆ(2

~

1

~1)1(

pexq

TB

n

SniipDp

X

nn

)1,...,1(|

),...,1,0(0|

0

0

DpixBX

pBXi

Xi

X

4. Black p-brane solution4. Black p-brane solution from boundary state from boundary state

pp

p

p

p

jiij

pp

rprGrGTrH

rHArHe

dxdxrHdxdxrHds

7)8(

1...01

4

3

8

1

8

72

)7(

1)(),(21)(

,)(,)(

,)()(

).(2

),(2

3

,)(8

1,)(

8

72

)1(...01

)1(

)1(

rGTA

rGTp

rGTp

rGTp

h

pp

p

ijpp

<B| |massless>

kmasslessge

BL

masslessmassless

BL

;0~.).

1

1

11

0

0

(e.g.) dilaton (10-dim.)(e.g.) dilaton (10-dim.)

)(21)7(

21)(

,)(

78

2

32ˆ222

rGTrp

TrH

rHee

ppp

p

p

)(22

3)(ˆ rGT

pr p

<B| 　 |φ> +…

We can extract each mode which are incluWe can extract each mode which are included in Φ, for example, dilaton, graviton, anded in Φ, for example, dilaton, graviton, antisym.tensor and so on. tisym.tensor and so on.

Such modes corresponds to the leading teSuch modes corresponds to the leading term of the classical solution.rm of the classical solution.

SFT action and source termSFT action and source term

tensionbraneDT

stateboundarysourceB

fieldantisymgravitondilaton

fieldstring

BTg

QS

p

p

:

)(:

,....).,,(

:

,32

1

Calculation of fieldsCalculation of fields

BBgTBT

eXdX

BTg

QS

pp

S

p

32

,32

1

<B| |> +<B|

<B||> +…

Here, we do not know how strings interact, Here, we do not know how strings interact, so we use 3-point coupling of SUGRA.so we use 3-point coupling of SUGRA.

SUGRASFT

(e.g.) dilaton (10-dim.)(e.g.) dilaton (10-dim.)

)(21)7(

21)(

,)(

78

2

32ˆ222

rGTrp

TrH

rHee

ppp

p

p

22 )(22

3)(

22

3)(ˆ rGT

prGT

pr pp

<B| 　 |φ> +<B|

<B| 　 |φ> +…

59

9

2

2

97

2

21429

22

22)2(

||)2()8(2

32

9

)5)(7(

3

22

1

)7(28

29

)3(

)(22

3ˆ

pi

xik

pi

p

pp

p

pp

p

p

k

ekd

p

pp

pp

pT

rp

Tp

p

rGTp

i

AAh

hg

Fp

pghRgxdI p

&

,2

,|ˆ|ˆ2

3exp

)!2(8

1ˆˆ2

1),(

2

1 222

10

ΦΦ ｈ A AΦ+

hμν

Φ

Φ

Φ

A

A

k1k1

k2k2

k3 k3

).()2(28

3)(

),()2(2

1)(

321)9(9

21...01...01

321)9(9

2121

kkkkkp

AA

kkkkkkkh

pppp

pp

,1

22ˆ

),,...,,,...,(1

2

,1

22

3ˆ

21...01

21

21

ippp

ipp

ipp

kVTAB

bbaadiagk

VThB

k

pVTB

kΦ

k

hμν

59

9

2

22

97

2

21429

22

22)2(

||)2()8(2

32

9

)5)(7(

3

)7(22

1

)7(28

29

)3(

)(22

3ˆ

pi

xik

pi

p

pp

p

pp

p

p

k

ekd

p

pp

pp

p

p

T

rp

Tp

p

rGTp

i

(c.f.) SUGRA(c.f.) SUGRA

JJTJTX pp32

Φhμν

Bμν

・・・

+ +…

5. Summary5. Summary

Black p-branes are the classical solutions Black p-branes are the classical solutions of SUGRA and they are thought to of SUGRA and they are thought to correspond to D-branes in low energy limit.correspond to D-branes in low energy limit.

Boundary states are another representation Boundary states are another representation of D-branes, which are written in closed of D-branes, which are written in closed string channel.string channel.

Using 3-point coupling of SUGRA, we can Using 3-point coupling of SUGRA, we can reproduce the asymptotic behavior of a reproduce the asymptotic behavior of a black p-brane from a boundary state .black p-brane from a boundary state .

6. Problem6. Problem

STF coupling ⇔ SUGRA coupling ?STF coupling ⇔ SUGRA coupling ?

Degrees of freedom of field-redefinitionDegrees of freedom of field-redefinitiongraviton of SFT ⇔ graviton of SUGRA ?graviton of SFT ⇔ graviton of SUGRA ?

Difference between D-brane Difference between D-brane and black p-brane and black p-brane 　　→ → massive mode effectmassive mode effect→ Hawking radiation, etc.→ Hawking radiation, etc.

7. Future Works7. Future Works We are now investigating...We are now investigating...

Classical solution for unstable non-BPS D-Classical solution for unstable non-BPS D-branebrane

D-brane deformation using String Field D-brane deformation using String Field Theory or CFTTheory or CFT

ApplicationApplication Hawking radiation in terms of D-braneHawking radiation in terms of D-brane D/anti-D brane inflationD/anti-D brane inflation